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2D S o l it o n s in Dipolar BECs

2D S o l it o n s in Dipolar BECs. 1 I. Tikhonenkov, 2 B. Malomed, and 1 A. Vardi 1 Department of Chemistry, Ben-Gurion University 2 Department of Physical Electronics, School of Electrical Engineering, Tel-Aviv University. Contact pseudopotential. Dilute Bose gas at low T.

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2D S o l it o n s in Dipolar BECs

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  1. 2D Solitons in Dipolar BECs 1I. Tikhonenkov, 2B. Malomed, and 1A. Vardi 1Department of Chemistry, Ben-Gurion University 2Department of Physical Electronics, School of Electrical Engineering, Tel-Aviv University Ein Gedi, Feb. 24-29, 2008

  2. Contact pseudopotential Dilute Bose gas at low T Ein Gedi, Feb. 24-29, 2008

  3. Gross-Pitaevskii description Condensate order-parameter • Lowest order mean-field theory: Gross-Pitaevskii energy functional: • minimize EGP under the constraint: Gross-Pitaevskii (nonlinear Schrödinger) equation: Ein Gedi, Feb. 24-29, 2008

  4. Variational Calculation • Evaluation of the EGP in an harmonic trap, using a gaussian solution with varying width b. • Kinetic energy per-particle varies as 1/b2 - dispersion. • Nonlinear interaction per-particle varies as gn- g/b3 in 3D, g/b in 1D. • In 1D with g<0, kinetic dispersion can balance attraction and arrest collapse. Ein Gedi, Feb. 24-29, 2008

  5. Solitons • Localized solutions of nonlinear differential equations. • Result in from the interplay of dispersive terms and nonlinear terms. • Propagate long distances without dispersion. • Collide without radiating. • Not affected by their excitations. Ein Gedi, Feb. 24-29, 2008

  6. Zero-temperature BEC solitons • NLSE in 1D with attractive interactions (g<0), no confinement Posesses self-localized sech soliton solutions: Bright soliton: Healing length at x=0 Chemical potential of a bright soliton Ein Gedi, Feb. 24-29, 2008

  7. Attractive interactions, (self-focusing nonlinearity) No interactions, matter wave dispersion time time x x Zero-temperature BEC solitons Ein Gedi, Feb. 24-29, 2008

  8. (3) Turn off both the trap and interactions (Feshbach mechanism) • Prepare BEC (static) • in the trap (2) Turn off the trap and let evolve L. Khaykovich et al. Science 296, 1290 (2002). Observation of BEC bright solitons Ein Gedi, Feb. 24-29, 2008

  9. Observation of BEC solitons Dark solitons by phase imprinting: J. Denschlag et al., Science 287, 5450 (2000). Bright solitons L. Khaykovich et al. Science 296, 1290 (2002). Bright soliton train: K. E. Strecker et al., Nature 417, 150 (2002). Ein Gedi, Feb. 24-29, 2008

  10. characteristic width of a 2D BEC wavefunction is monotonic in collapse expansion Instability of 2D solitons without dipolar-interaction Ein Gedi, Feb. 24-29, 2008

  11. Dipole-dipole interaction vacuumpermittivity d - magnetic/electric dipole moment Ein Gedi, Feb. 24-29, 2008

  12. Units Ein Gedi, Feb. 24-29, 2008

  13. 2D Bright solitons in dipolar BECs P. Pedri and L. Santos, PRL 95, 200404 (2005) Ein Gedi, Feb. 24-29, 2008

  14. The total dipolar interaction is attractive at L< Lzand repulsive at L> Lz. There is a maximum in E(L, hence no soliton. Manipulation of dipole-dipole interaction • In order to stabilize 2D solitary waves in the PS configuration, it is necessary to reversedipole-dipole behavior, so that side-by-side dipoles attract each other and head-to-tail dipoles repell one another. Ein Gedi, Feb. 24-29, 2008

  15. Manipulation of dipole-dipole interactionS. Giovanazzi, A. Goerlitz, and T. Pfau, PRL 89, 130401 (2002) • The magnetic dipole interaction can be tuned, using rotating fields from +Vdat , to -Vd/2at  • The maximum becomes a minimum and 2D bright SWs can be found, provided that the dipole term is sufficiently strong to overcome the kinetic+contact terms, i.e. • Or, for Ein Gedi, Feb. 24-29, 2008

  16. E for confinement along the dipolar axis z, gaussian ansatz, g=500 Ein Gedi, Feb. 24-29, 2008

  17. Dipolar axis in the 2D planeI. Tikhonenkov, B. A. Malomed, and AV, PRL 100, 090406 (2008) Ein Gedi, Feb. 24-29, 2008

  18. y z x Dipolar axis in the 2D plane For gd > 0 stable self trapping along the dipolar axis z: Ein Gedi, Feb. 24-29, 2008

  19. y y z Self trapping along x is enabled by the interplay of 1/Lx2 kinetic dispersion and -1/Lx dipolar attraction z x x For gd > 0, what happens along x ? Ein Gedi, Feb. 24-29, 2008

  20. E for confinement perpendicular to the dipolar axis Ein Gedi, Feb. 24-29, 2008

  21. 3D Propagation and stability Ein Gedi, Feb. 24-29, 2008

  22. Deviation from /2 rotated soliton at t= /2 Driven Rotation Ein Gedi, Feb. 24-29, 2008

  23. Experimental realization • 52Cr (magnetic dipole moment d=6B) • Dipolar molecules (electric dipole of ~0.1-1D) For g,gd > 0 : Ein Gedi, Feb. 24-29, 2008

  24. Conclusions • 2D bright solitons exist for dipolar alignment in the free-motion plane. • For this configuration, no special tayloring of dipole-dipole interactions is called for. • The resulting solitary waves are unisotropic in the 2D plane, hence interesting soliton collision dynamics. Ein Gedi, Feb. 24-29, 2008

  25. Incoherent matter-wave Solitons 1,2H. Buljan, 1M. Segev, and 3A. Vardi 1Department of Physics, The Technion 2Department of Physics, Zagreb Univesity 3Department of Chemistry, Ben-Gurion University Ein Gedi, Feb. 24-29, 2008

  26. What about quantum/thermal fluctuations ? Trap OFF → nonequilibrium dynamics Prepared (static) BEC partially condensed Condensed particles ? Thermal cloud BEC-soliton dynamics affected by (1) Thermal cloud (and vice versa) (2) Condensate depletion during dynamics Ein Gedi, Feb. 24-29, 2008

  27. condensate fluctuations T=0 - Bogoliubov theory (ask Nir) • Want to calculate zero temperature fluctuations. • Separate: • retain quadratic fluctuation terms and add N0 constraint: Ein Gedi, Feb. 24-29, 2008

  28. v(x) T=0 - Bogoliubov theory • Bogoliubov transformation: Ein Gedi, Feb. 24-29, 2008

  29. Bogoliubov spectrum of a bright soliton • linearize about a bright soliton solution:

  30. Bogoliubov spectrum of a bright solitonScattering without reflection • Transmittance: • Bogoliubov quasiparticles scatter without reflection on • the soliton (B. Eiermann et al., PRL 92, 230401 (2004), • S. Sinha et al., PRL 96, 030406 (2006)). Ein Gedi, Feb. 24-29, 2008

  31. no exchange ! Limitations on Bogoliubov theory • The condensate number is fixed - no backreaction • The GP energy is treated separately from the fluctuations pair production direct + exchange Due to exchange energy in collisions between condensate particles and excitations, it may be possible to gain energy By exciting pairs of particles from the condensate ! Ein Gedi, Feb. 24-29, 2008

  32. Heisenberg eq. of motion for the Bose field operator Fluctuations Condensate TDHFB approximation • separate, like before • retain quadratic terms in the fluctuations, to obtain coupled equations for: Condensate order-parameter Pair correlation functions - single particle normal and anomalous densities

  33. Condensate density Normal noncondensate density Anomalous noncondensate density TDHFB approximation (e.g., Proukakis, Burnett, J. Res. NIST 1996, Holland et al., PRL 86 (2001)) Ein Gedi, Feb. 24-29, 2008

  34. Bose distribution Initial Conditions - static HFB solution in a trap Fluctuations do not vanish even at T=0, quantum fluct. Ein Gedi, Feb. 24-29, 2008

  35. Initial conditions: Dynamics - TDHFB equations Ein Gedi, Feb. 24-29, 2008

  36. System Parameters  • Quasi 1D geometry x • Parameters close to experiment: • N = 2.2 1047Li atoms • ω= 4907 Hz ; a = 1.3 μm • ωx = 439 Hz ; ax = 4.5 μm • Na3D = -0.68 μm • TDHFB can be used only for limited time-scales: • Tevolutionω <<Tcollisionalω~ 104 Ein Gedi, Feb. 24-29, 2008

  37. TDHFB vs. GP Instability pairing GPE evolution, mechanical stability PRL 80, 180401 (2005) TDHFB: Without interactions matter wave dispersion Dynamical condensate depletion

  38. Incoherent matter-wave solitons Correlations Mixture of condensed and noncondensed atoms Re μ(x1,x2,t=0) Re μ(x1,x2,t) Im μ(x1,x2,t) Ein Gedi, Feb. 24-29, 2008

  39. Number and energy conservation Number conservation Energy conservation condensate kinetic energy thermal cloud kinetic energy total interaction energy condensate fraction thermal population Ein Gedi, Feb. 24-29, 2008

  40. Conclusions • Dynamics of a partially condensed Bose gas calculated via a nonlinear TDHFB model • Noncondensed particles (thermal/quantum) affect the dynamics of BEC solitons • Pairing instability - dynamical depletion of a BEC with attractive interactions • Incoherent matter-wave solitons constituting both condensed and noncondensed particles • Analogy with optics: Coherent light in Kerr media Ξ zero-temperature BEC Partially (in)coherent light in Kerr media Ξ partially condensed BEC Ein Gedi, Feb. 24-29, 2008

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